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| Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for submateq 33808. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
| Ref | Expression |
|---|---|
| submateqlem2.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| submateqlem2.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) |
| submateqlem2.m | ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) |
| submateqlem2.1 | ⊢ (𝜑 → 𝑀 < 𝐾) |
| Ref | Expression |
|---|---|
| submateqlem2 | ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fz1ssnn 13595 | . . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
| 2 | submateqlem2.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
| 3 | 1, 2 | sselid 3981 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 4 | 3 | nnge1d 12314 | . . . 4 ⊢ (𝜑 → 1 ≤ 𝑀) |
| 5 | submateqlem2.1 | . . . 4 ⊢ (𝜑 → 𝑀 < 𝐾) | |
| 6 | 4, 5 | jca 511 | . . 3 ⊢ (𝜑 → (1 ≤ 𝑀 ∧ 𝑀 < 𝐾)) |
| 7 | 2 | elfzelzd 13565 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 8 | 1zzd 12648 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 9 | submateqlem2.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
| 10 | 9 | elfzelzd 13565 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
| 11 | elfzo 13701 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) | |
| 12 | 7, 8, 10, 11 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) |
| 13 | 6, 12 | mpbird 257 | . 2 ⊢ (𝜑 → 𝑀 ∈ (1..^𝐾)) |
| 14 | 2 | orcd 874 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁)) |
| 15 | submateqlem2.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 16 | nnuz 12921 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
| 17 | 15, 16 | eleqtrdi 2851 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1)) |
| 18 | fzm1 13647 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) | |
| 19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) |
| 20 | 14, 19 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
| 21 | 3 | nnred 12281 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 22 | 21, 5 | ltned 11397 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐾) |
| 23 | nelsn 4666 | . . . 4 ⊢ (𝑀 ≠ 𝐾 → ¬ 𝑀 ∈ {𝐾}) | |
| 24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝑀 ∈ {𝐾}) |
| 25 | 20, 24 | eldifd 3962 | . 2 ⊢ (𝜑 → 𝑀 ∈ ((1...𝑁) ∖ {𝐾})) |
| 26 | 13, 25 | jca 511 | 1 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 1c1 11156 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 ..^cfzo 13694 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 |
| This theorem is referenced by: submateq 33808 |
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